Advances in Risk Management

84 MANAGING INTEREST RATE RISK UNDER NON-PARALLEL CHANGES

based on a continuous-time model for interest rates. This model assumes
that default free discount bond prices are determined by the time to maturity
and two factors, the long-term interest rate and the spread.
The generalized duration is useful to compute hedging ratios. We have
alsoshownanumericalexamplethatillustrateshowthesenewmeasurescan
mitigate the limitations of the conventional duration. Analysing different
situations, it has been checked that the generalized durations do provide
adequate information on the future behavior of a bond portfolio with respect
to unexpected changes in the yield curve.
Hence, these measures can be a useful tool for managing fixed income
portfolios. The relevant characteristics to determine the future behavior of
these portfolios are (a) the generalized duration with respect to the short-
term interest rate and (b) the expectations on the movements (increase or
decrease) in interest rates. Thus, if two portfolios have the same generalized
durations with respect to the spread and to the long-term rate, the best
portfolio is the one with lower (higher) generalized duration with respect
to the short-term interest rate if interest rates rise (fall).

NOTES

1. This fact is illustrated in Nelson and Schaefer (1983) and Smithson and Smith (1995).


2. Jones (1991), Litterman and Scheinkman (1991), and Knez, Litterman and Scheikman
   (1994) show empirical evidence of these movements.


3. See for instance, Ingersoll, Skelton and Weil (1978), Cox, Ingersoll and Ross (1979),
   and D’Ecclesia and Zenios (1994), among others.


4. These limitations are because conventional duration does not provide adequate infor-
   mation about the future performance of a bond portfolio when the yield curve changes
   in a non-parallel way. That is, the relative behavior of two portfolios with the same
   modified duration, measured by the difference in yields, depends on the size and the
   type of change in yields.


5. This assumption has been empirically shown in papers as Ayres and Barry (1980),
   Schaefer (1980) and Nelson and Schaefer (1983), among others.


6. Many other types of interest rates derivatives were priced by solving the valuation
   equation with the appropriate terminal condition. See Moreno (2003) for details.


7. This value is an indicative measure of the change in the call price to changes in this
   factor.


8. See Fabozzi (1993), chapter 15.

REFERENCES

Ayres, H.R. and Barry, J.Y. (1980) “A Theory of the U.S. Treasury Market Equilibrium”,
Management Science, 26(6): 539–69.
Bierwag, G.O., Kaufman, G.G. and Toevs, A. (1983)Innovations in Bond Portfolio
Management: Duration Analysis and Immunization(Greenwich, CT: JAI Press).
Chen, L. (1996)Interest Rate Dynamics, Derivatives Pricing, and Risk Management(Berlin:
Springer-Verlag).